Reformat examples

diff --git a/examples/bac2014.py b/examples/bac2014.py
index 775be66..cc02126 100755 (executable)
--- a/examples/bac2014.py
+++ b/examples/bac2014.py
@@ -1,9 +1,15 @@
 #!/usr/bin/env python3
 
 #!/usr/bin/env python3
 

+# This example is inspired from a math question in the French baccalauréat 2014,
+# consisting in computing the intersection of a plane with a line.
+

 from linpy import *
 
 x, y, z = symbols('x y z')
 from linpy import *
 
 x, y, z = symbols('x y z')

-DF = Eq(x, y) & Eq(z, 6 - 2*x)
-P = Eq(x + y - 2*z, 0)
+plane = Eq(x, y) & Eq(z, 6 - 2*x)
+line = Eq(x + y - 2*z, 0)

 
 

-print('DF ∩ P =', DF & P)
+if __name__ == '__main__':
+    print('plane:       ', plane)
+    print('line:        ', line)
+    print('intersection:', plane & line)

diff --git a/examples/diamonds.py b/examples/diamonds.py
deleted file mode 100755 (executable)
index 0978d4c..0000000
--- a/examples/diamonds.py
+++ /dev/null
@@ -1,48 +0,0 @@
-#!/usr/bin/env python3
-
-import matplotlib.pyplot as plt
-
-from matplotlib import pylab
-from mpl_toolkits.mplot3d import Axes3D
-
-from linpy import *
-
-x, y, z = symbols('x y z')
-
-fig = plt.figure(facecolor='white')
-
-diam_plot = fig.add_subplot(2, 2, 1, aspect='equal')
-diam_plot.set_title('Diamond')
-diam = Ge(y, x - 1) & Le(y, x + 1) & Ge(y, -x - 1) & Le(y, -x + 1)
-diam.plot(diam_plot, fill=True, edgecolor='red', facecolor='yellow')
-
-cham_plot = fig.add_subplot(2, 2, 2, projection='3d', aspect='equal')
-cham_plot.set_title('Chamfered cube')
-cham = Le(0, x) & Le(x, 3) & Le(0, y) & Le(y, 3) & Le(0, z) & Le(z, 3) & \
-    Le(z - 2, x) & Le(x, z + 2) & Le(1 - z, x) & Le(x, 5 - z) & \
-    Le(z - 2, y) & Le(y, z + 2) & Le(1 - z, y) & Le(y, 5 - z) & \
-    Le(y - 2, x) & Le(x, y + 2) & Le(1 - y, x) & Le(x, 5 - y)
-cham.plot(cham_plot, facecolors=(1, 0, 0, 0.75))
-
-rhom_plot = fig.add_subplot(2, 2, 3, projection='3d', aspect='equal')
-rhom_plot.set_title('Rhombicuboctahedron')
-rhom = cham & \
-    Le(x + y + z, 7) & Ge(-2, -x - y - z) & \
-    Le(-1, x + y - z) & Le(x + y - z, 4) & \
-    Le(-1, x - y + z) & Le(x - y + z, 4) & \
-    Le(-1, -x + y + z) & Le(-x + y + z, 4)
-rhom.plot(rhom_plot, facecolors=(0, 1, 0, 0.75))
-
-cubo_plot = fig.add_subplot(2, 2, 4, projection='3d', aspect='equal')
-cubo_plot.set_title('Truncated cuboctahedron')
-cubo = Le(0, x) & Le(x, 5) & Le(0, y) & Le(y, 5) & Le(0, z) & Le(z, 5) & \
-    Le(x -4, y) & Le(y, x + 4) & Le(-x + 1, y) & Le(y, -x + 9) & \
-    Le(y -4, z) & Le(z, y + 4) & Le(-y + 1, z) & Le(z, -y + 9) & \
-    Le(z -4, x) & Le(x, z + 4) & Le(-z + 1, x) & Le(x, -z + 9) & \
-    Le(3, x + y + z) & Le(x + y + z, 12) & \
-    Le(-2, x - y + z) & Le(x - y + z, 7) & \
-    Le(-2, -x + y + z) & Le(-x + y + z, 7) & \
-    Le(-2, x + y - z) & Le(x + y - z, 7)
-cubo.plot(cubo_plot, facecolors=(0, 0, 1, 0.75))
-
-pylab.show()

diff --git a/examples/menger.py b/examples/menger.py
index d8a74d4..dcba976 100755 (executable)
--- a/examples/menger.py
+++ b/examples/menger.py
@@ -1,5 +1,18 @@
 #!/usr/bin/env python3
 
 #!/usr/bin/env python3
 

+# Plot a Menger sponge.
+#
+# The construction of a Menger sponge can be described as follows:
+#
+# 1. Begin with a cube.
+# 2. Divide every face of the cube into 9 squares, like a Rubik's Cube. This
+#    will sub-divide the cube into 27 smaller cubes.
+# 3. Remove the smaller cube in the middle of each face, and remove the smaller
+#    cube in the very center of the larger cube, leaving 20 smaller cubes. This
+#    is a level-1 Menger sponge (resembling a Void Cube).
+# 4. Repeat steps 2 and 3 for each of the remaining smaller cubes, and continue
+#    to iterate.
+

 import argparse
 
 import matplotlib.pyplot as plt
 import argparse
 
 import matplotlib.pyplot as plt


@@ -11,6 +24,7 @@ from mpl_toolkits.mplot3d import Axes3D
 
 from linpy import *
 
 
 from linpy import *
 

+

 x, y, z = symbols('x y z')
 
 _x, _y, _z = x.asdummy(), y.asdummy(), z.asdummy()
 x, y, z = symbols('x y z')
 
 _x, _y, _z = x.asdummy(), y.asdummy(), z.asdummy()


@@ -56,8 +70,8 @@ def menger(domain, count=1, cut=False):
 if __name__ == '__main__':
     parser = argparse.ArgumentParser(
         description='Compute a Menger sponge.')
 if __name__ == '__main__':
     parser = argparse.ArgumentParser(
         description='Compute a Menger sponge.')

-    parser.add_argument('-n', '--iterations', type=int, default=1,
-        help='number of iterations (default: 1)')
+    parser.add_argument('-n', '--iterations', type=int, default=2,
+        help='number of iterations (default: 2)')

     parser.add_argument('-c', '--cut', action='store_true', default=False,
         help='cut the sponge')
     args = parser.parse_args()
     parser.add_argument('-c', '--cut', action='store_true', default=False,
         help='cut the sponge')
     args = parser.parse_args()

diff --git a/examples/nsad2010.py b/examples/nsad2010.py
index 9359315..4b73eef 100755 (executable)
--- a/examples/nsad2010.py
+++ b/examples/nsad2010.py
@@ -1,5 +1,13 @@
 #!/usr/bin/env python3
 
 #!/usr/bin/env python3
 

+# This is an implementation of the algorithm described in
+#
+# [ACI10] C. Ancourt, F. Coelho and F. Irigoin, A modular static analysis
+# approach to affine loop invariants detection (2010), pp. 3 - 16, NSAD 2010.
+#
+# to compute the transitive closure of an affine transformer. A refined version
+# of this algorithm is implemented in PIPS.
+

 from linpy import *
 
 
 from linpy import *
 
 


@@ -39,8 +47,8 @@ class Transformer:
 
 
 if __name__ == '__main__':
 
 
 if __name__ == '__main__':

-    i, iprime, j, jprime = symbols("i i' j j'")
-    transformer = Transformer(Eq(iprime, i + 2) & Eq(jprime, j + 1),
-        [i, j], [iprime, jprime])
+    i0, i, j0, j = symbols('i0 i j0 j')
+    transformer = Transformer(Eq(i, i0 + 2) & Eq(j, j0 + 1),
+        [i0, j0], [i, j])

     print('T  =', transformer.polyhedron)
     print('T* =', transformer.star().polyhedron)
     print('T  =', transformer.polyhedron)
     print('T* =', transformer.star().polyhedron)

diff --git a/examples/plots.py b/examples/plots.py
new file mode 100755 (executable)
index 0000000..d172835
--- /dev/null
+++ b/examples/plots.py
@@ -0,0 +1,55 @@
+#!/usr/bin/env python3
+
+# This program plots several 2D and 3D polyhedra on the same figure,
+# illustrating some of the possible plot options.
+
+import matplotlib.pyplot as plt
+
+from matplotlib import pylab
+from mpl_toolkits.mplot3d import Axes3D
+
+from linpy import *
+
+
+x, y, z = symbols('x y z')
+
+diam = Ge(y, x - 1) & Le(y, x + 1) & Ge(y, -x - 1) & Le(y, -x + 1)
+
+cham = Le(0, x, 3) & Le(0, y, 3) & Le(0, z, 3) & \
+    Le(z - 2, x, z + 2) & Le(1 - z, x, 5 - z) & \
+    Le(z - 2, y, z + 2) & Le(1 - z, y, 5 - z) & \
+    Le(y - 2, x, y + 2) & Le(1 - y, x, 5 - y)
+
+rhom = cham & \
+    Le(x + y + z, 7) & Ge(-2, -x - y - z) & \
+    Le(-1, x + y - z, 4) & Le(-1, x - y + z, 4) & Le(-1, -x + y + z, 4)
+
+cubo = Le(0, x, 5) & Le(0, y, 5) & Le(0, z, 5) & \
+    Le(x -4, y, x + 4) & Le(-x + 1, y, -x + 9) & \
+    Le(y -4, z, y + 4) & Le(-y + 1, z, -y + 9) & \
+    Le(z -4, x, z + 4) & Le(-z + 1, x, -z + 9) & \
+    Le(3, x + y + z, 12) & Le(-2, x - y + z, 7) & \
+    Le(-2, -x + y + z, 7) & Le(-2, x + y - z, 7)
+
+
+if __name__ == '__main__':
+
+    fig = plt.figure(facecolor='white')
+
+    diam_plot = fig.add_subplot(2, 2, 1, aspect='equal')
+    diam_plot.set_title('Diamond')
+    diam.plot(diam_plot, fill=True, edgecolor='red', facecolor='yellow')
+
+    cham_plot = fig.add_subplot(2, 2, 2, projection='3d', aspect='equal')
+    cham_plot.set_title('Chamfered cube')
+    cham.plot(cham_plot, facecolors=(1, 0, 0, 0.75))
+
+    rhom_plot = fig.add_subplot(2, 2, 3, projection='3d', aspect='equal')
+    rhom_plot.set_title('Rhombicuboctahedron')
+    rhom.plot(rhom_plot, facecolors=(0, 1, 0, 0.75))
+
+    cubo_plot = fig.add_subplot(2, 2, 4, projection='3d', aspect='equal')
+    cubo_plot.set_title('Truncated cuboctahedron')
+    cubo.plot(cubo_plot, facecolors=(0, 0, 1, 0.75))
+
+    pylab.show()

diff --git a/examples/squares.py b/examples/squares.py
index 98e2ca8..15aed16 100755 (executable)
--- a/examples/squares.py
+++ b/examples/squares.py
@@ -1,87 +1,56 @@
 #!/usr/bin/env python3
 
 #!/usr/bin/env python3
 

-from linpy import *
-import matplotlib.pyplot as plt
-from matplotlib import pylab
-
-a, x, y, z = symbols('a x y z')
-
-sq1 = Le(0, x) & Le(x, 2) & Le(0, y) & Le(y, 2)
-sq2 = Le(2, x) & Le(x, 4) & Le(2, y) & Le(y, 4)
-sq3 = Le(0, x) & Le(x, 3) & Le(0, y) & Le(y, 3)
-sq4 = Le(1, x) & Le(x, 2) & Le(1, y) & Le(y, 2)
-sq5 = Le(1, x) & Le(x, 2) & Le(1, y)
-sq6 = Le(1, x) & Le(x, 2) & Le(1, y) & Le(y, 3)
-sq7 = Le(0, x) & Le(x, 2) & Le(0, y) & Eq(z, 2) & Le(a, 3)
-p = Le(2*x+1, y) & Le(-2*x-1, y) & Le(y, 1)
-
-universe = Polyhedron([])
-q = sq1 - sq2
-e = Empty
-
-print('sq1 =', sq1) #print correct square
-print('sq2 =', sq2) #print correct square
-print('sq3 =', sq3) #print correct square
-print('sq4 =', sq4) #print correct square
-print('universe =', universe) #print correct square
-print()
-print('¬sq1 =', ~sq1) #test complement
-print()
-print('sq1 + sq1 =', sq1 + sq2) #test addition
-print('sq1 + sq2 =', Polyhedron(sq1 + sq2)) #test addition
-print()
-print('universe + universe =', universe + universe)#test addition
-print('universe - universe =', universe - universe) #test subtraction
-print()
-print('sq2 - sq1 =', sq2 - sq1) #test subtraction
-print('sq2 - sq1 =', Polyhedron(sq2 - sq1)) #test subtraction
-print('sq1 - sq1 =', Polyhedron(sq1 - sq1)) #test subtraction
-print()
-print('sq1 ∩ sq2 =', sq1 & sq2) #test intersection
-print('sq1 ∪ sq2 =', sq1 | sq2) #test union
-print()
-print('sq1 ⊔ sq2 =', Polyhedron(sq1 | sq2)) # test convex union
-print()
-print('check if sq1 and sq2 disjoint:', sq1.isdisjoint(sq2)) #should return false
-print()
-print('sq1 disjoint:', sq1.disjoint()) #make disjoint
-print('sq2 disjoint:', sq2.disjoint()) #make disjoint
-print()
-print('is square 1 universe?:', sq1.isuniverse()) #test if square is universe
-print('is u universe?:', universe.isuniverse()) #test if square is universe
-print()
-print('is sq1 a subset of sq2?:', sq1.issubset(sq2)) #test issubset()
-print('is sq4 less than sq3?:', sq4.__lt__(sq3)) # test lt(), must be a strict subset
-print()
-print('lexographic min of sq1:', sq1.lexmin()) #test lexmin()
-print('lexographic max of sq1:', sq1.lexmax()) #test lexmin()
-print()
-print('lexographic min of sq2:', sq2.lexmin()) #test lexmax()
-print('lexographic max of sq2:', sq2.lexmax()) #test lexmax()
-print()
-print('Polyhedral hull of sq1 + sq2 is:', q.aspolyhedron()) #test polyhedral hull
-print()
-print('is sq1 bounded?', sq1.isbounded()) #bounded should return True
-print('is sq5 bounded?', sq5.isbounded()) #unbounded should return False
-print()
-print('sq6:', sq6)
-print('sample Polyhedron from sq6:', sq6.sample())
-print()
-print('sq7 with out constraints involving y and a', sq7.project([a, z, x, y])) 
-print()
-print('the verticies for s are:', p.vertices())
-
-
-# plotting the intersection of two squares
-square1 = Le(0, x) & Le(x, 2) & Le(0, y) & Le(y, 2)
-square2 = Le(1, x) & Le(x, 3) & Le(1, y) & Le(y, 3)
-
-fig = plt.figure()
-plot = fig.add_subplot(1, 1, 1, aspect='equal')
-square1.plot(plot, facecolor='red', alpha=0.3)
-square2.plot(plot, facecolor='blue', alpha=0.3)
-
-squares = Polyhedron(square1 + square2)
-squares.plot(plot, facecolor='blue', alpha=0.3)
-
-pylab.show()
+# This is the code example used in the tutorial. It shows how to define and
+# manipulate polyhedra.
+
+import code
+
+
+class InteractiveConsole(code.InteractiveConsole):
+    def push(self, line=''):
+        if line:
+            print('>>>', line)
+            return super().push(line)
+        else:
+            print()
+
+
+if __name__ == '__main__':
+
+    shell = InteractiveConsole()
+
+    shell.push('from linpy import *')
+    shell.push("x, y = symbols('x y')")
+    shell.push()
+
+    shell.push('square1 = Le(0, x, 2) & Le(0, y, 2)')
+    shell.push('square1')
+    shell.push()
+
+    shell.push("square2 = Polyhedron('1 <= x <= 3, 1 <= y <= 3')")
+    shell.push('square2')
+    shell.push()
+
+    shell.push('inter = square1.intersection(square2)')
+    shell.push('inter')
+    shell.push()
+
+    shell.push('hull = square1.convex_union(square2)')
+    shell.push('hull')
+    shell.push()
+
+    shell.push('square1.project([y])')
+    shell.push()
+
+    shell.push('inter <= square1')
+    shell.push('inter == Empty')
+    shell.push()
+
+    shell.push('union = square1 | square2')
+    shell.push('union')
+    shell.push('union <= hull')
+    shell.push()
+
+    shell.push('diff = square1 - square2')
+    shell.push('diff')
+    shell.push('~square1')

diff --git a/examples/tesseract.py b/examples/tesseract.py
index 0a57188..383d7bd 100755 (executable)
--- a/examples/tesseract.py
+++ b/examples/tesseract.py
@@ -1,23 +1,25 @@
 #!/usr/bin/env python3
 
 #!/usr/bin/env python3
 

+# In geometry, the tesseract is the four-dimensional analog of the cube; the
+# tesseract is to the cube as the cube is to the square. Just as the surface of
+# the cube consists of 6 square faces, the hypersurface of the tesseract
+# consists of 8 cubical cells.
+

 from linpy import *
 
 from linpy import *
 

+

 x, y, z, t = symbols('x y z t')
 
 x, y, z, t = symbols('x y z t')
 

-tesseract = \
-    Le(0, x) & Le(x, 1) & \
-    Le(0, y) & Le(y, 1) & \
-    Le(0, z) & Le(z, 1) & \
-    Le(0, t) & Le(t, 1)
+tesseract = Le(0, x, 1) & Le(0, y, 1) & Le(0, z, 1) & Le(0, t, 1)

 
 def faces(polyhedron):
     for points in polyhedron.faces():
         face = points[0].aspolyhedron()
 
 def faces(polyhedron):
     for points in polyhedron.faces():
         face = points[0].aspolyhedron()

-        face = face.union(*[point.aspolyhedron() for point in points[1:]])
-        face = face.aspolyhedron()
+        face = face.convex_union(*[point.aspolyhedron() for point in points[1:]])

         yield face
 
         yield face
 

-print('Faces of tesseract\n\n  {}\n\nare:\n'.format(tesseract))
-for face in faces(tesseract):
-    assert(len(face.vertices()) == 8)
-    print('  {}'.format(face))
+if __name__ == '__main__':
+    print('Faces of tesseract\n\n  {}\n\nare:\n'.format(tesseract))
+    for face in faces(tesseract):
+        assert(len(face.vertices()) == 8)
+        print('  {}'.format(face))